Home > Optimization Methods in Computational Fluid Dynamics

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In order to set the context for aerodynamic shape opti- mization, the airplane design process is summarized in the following paragraphs. It can generally be divided into three phases: conceptual design, preliminary design, and final detailed design, as illustrated in Figure 1. The conceptual design stage defines the mission in the light of anticipated market requirements and determines a general preliminary configuration capable of performing this mission, together

with first estimates of size, weight, and performance. In the preliminary design stage, the aerodynamic shape and structural skeleton progress to the point where detailed per- formance estimates can be made and guaranteed to potential customers. At this stage, the development costs are still fairly moderate, in the range of 50–100 million dollars. In the final design stage, the structure must be defined in complete detail, together with complete systems, including the flight deck,

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control systems (involving major software development for fly-by-wire systems), electrical and hydraulic systems, land- ing gear, weapon systems for military aircraft, and cabin layout for commercial aircraft. Major costs are incurred at this stage, during which it is also necessary to prepare a detailed manufacturing plan, together with appropriate facilities and tooling. The development costs to reach the point of initial production are in the range of 3–10 billion dollars. Thus, the final design would normally be carried out only if suffi- cient orders have been received to indicate a reasonably high probability of recovering the investment. In the development of commercial aircraft, aerodynamic design plays a leading role in the preliminary design stage. The definition of the external aerodynamic shape may actu- ally be finalized in the preliminary design. The aerodynamic lines of the Boeing 777 were frozen, for example, when ini- tial orders were accepted before the initiation of the detailed design of the structure. Figure 2 illustrates the way in which the aerodynamic design process is embedded in the overall preliminary design. The starting point is an initial Com- puter Aided Design (CAD) definition resulting from the conceptual design. The inner loop of aerodynamic analysis is contained in an outer multi-disciplinary loop, which is in turn contained in a major design cycle involving wind tunnel testing. In recent Boeing practice, three major design cycles, each requiring about 4–6 months, have been used to finalize the wing design.

The use of computational simulation to scan many alterna- tive designs has proved extremely valuable in practice, but it is also evident that the number of possible design variations is too large to permit their exhaustive evaluation, and thus it is very unlikely that a truly optimum solution can be found without the assistance of automatic optimization procedures. To ensure the realization of the true best design, the ulti- mate goal of computational simulation methods should not just be the analysis of prescribed shapes but the automatic determination of the true optimum shape for the intended application. The need to find optimum aerodynamic designs was already well recognized by the pioneers of classical aerody- namic theory. A notable example is the determination that the optimum span-load distribution that minimizes the induced drag of a monoplane wing is elliptic (Glauert, 1926; Prandtl and Tietjens, 1934). There are also a number of famous results for linearized supersonic flow. The body of revolution of min- imum drag was determined by Sears (1947), while conditions for minimum drag of thin wings due to thickness and sweep were derived by Jones (1981). The problem of designing a two-dimensional profile to attain a desired pressure distribu- tion was studied by Lighthill (1945), who solved it for the case of incompressible flow with a conformal mapping of the profile to a unit circle. The speed over the profile is

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The simplest approach to optimization is to define the geom- etry through a set of design parameters, which may, for example, be the weights

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Consider the case of two-dimensional compressible inviscid flow. In the absence of shock waves, an initially irrotational flow will remain irrotational, and we can assume that the velocity vector

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The application of control theory to aerodynamic design problems is illustrated in this section for the case of three- dimensional wing design using the compressible Euler equations as the mathematical model. The extension of the method to treat the Navier–Stokes equations is presented in references Jameson, Martinelli and Pierce (1998), Jameson and Martinelli (1999), and Jameson (2003). It proves con- venient to denote the Cartesian coordinates and velocity

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When the inviscid Euler equations are used to model the flow, the source of drag is the wave drag due to shock waves. Accordingly, if the shape is optimized for minimum drag at fixed lift, the best attainable result is a shock-free airfoil with zero drag. By this criterion, the optimum shape is com- pletely non-unique, since all shock-free profiles are equally good. Experience during the last 15 years has confirmed that shock-free profiles can be obtained from a wide variety of initial shape, while maintaining a fixed lift coefficient and a fixed thickness. An example of shock-free design is shown in Figure 4 for the optimization of a DLBA-243 airfoil (Harbeck and Jameson, 2005).

Using three-dimensional optimization, it is possible to arrive very rapidly at an efficient design. This is illustrated in the next example that is the wing design for a proposed

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propeller-driven Reno Air Racer, the Shark, which was intended to reach speeds of 550 mph (Jameson and Vassberg, 2001). A sketch of Shark is shown in Figure 5a. The initial wing had extremely strong shock waves that were removed by optimization using the Euler equations, as shown in Figure 5b and 5c. Then the design was fur- ther refined by inverse design with the Reynolds-averaged Navier–Stokes equations to produce a pressure distribution that could promote laminar flow over a range of lift coeffi- cients as shown in Figure 5d.

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0.5E+00 0.6E+00 0.0E+00 0.8E+01 0.2E+02 0.2E+02 0.3E+02

0.4E+02 0.5E+02 0.00 50.00 100.00 150.00 200.00 250.00 300.00

280 260 240 220 200 180 160 140 120 100 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92

Baseline Redesign

In order to obtain the maximum possible benefit, one should allow for redesign of the planform as well as the wing sec- tion shapes. Then, it is necessary to estimate the change in the wing weight that will result from the redesign. The next example, a new wing design for a ��Super Boeing 747��, with an increased span, reduced sweep, and a much thicker wing section to reduce the weight, illustrates the kind of improve- ments that can be obtained by optimization. The numbers in Figure 6a show, while the curve in Figure 6b compares the drag rise of the ��Super B747�� with the existing B747 (Leoviriyakit and Jameson, 2004).

The same design method has also been applied to several complete aircraft configurations using unstructured meshes (Jameson, Shankaran and Martinelli, 2008). The results for a business jet are shown in Figure 7a and 7b. There is a

strong shock over the outboard wing sections of the initial configuration, which is essentially eliminated by the redesign. The drag was reduced from 235 counts to 215 counts in about eight design cycles. The lift was constrained at 0.4 by per- turbing the angle of attack. Further, the original thickness of the wing was maintained during the design process ensuring that fuel volume and structural integrity will be maintained by the redesigned shape. Thickness constraints on the wing were imposed on cutting planes along the span of the wing and by transferring the constrained shape movement back to the nodes of the surface triangulation.

The adjoint design method presented in these notes is now well established and has proved effective in a variety of indus- trial applications including, most recently, the wing design of the Gulfstream G650. The method combines the versatility of numerical optimization methods with the efficiency of inverse design methods. The geometry is modified by a

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The author��s research on shape optimization has benefited greatly from the continuing support over more than 20 years of the Air Force Office of Scientific Research through its Computational Mathematics Program.

Anderson, W.K. and Venkatakrishnan, V. (1997) Aerodynamic design and optimization on unstructured grids with a continuous adjoint formulation. AIAA 35th Aerospace Sciences Meeting, Reno. AIAA Paper 97-0643. Baysal, O. and Eleshaky, M.E. (1991) Aerodynamic design opti- mization using sensitivity analysis and computational fluid dynamics. 29th Aerospace Sciences Meeting, Reno. AIAA Paper 91-0471. Cabuk,H., Shung, C.H. and Modi, V. (1991) Adjoint operator approach to shape design for internal incompressible flow. Third International Conference on Inverse Design Concepts and Opti- mization in Engineering Sciences, Pennsylvania State University, pp. 391–404. Castro, C. and Zuazua, E. (2007) Systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids.

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Mavriplis, D.J. (2007) A discrete adjoint based approach for opti- mization problems on three-dimensional unstructured meshes.

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